Projection pair
A projection pair is a pair of two rational intervals which can be employed by the Scala "project" command to reduce a JI scale to a scale in a JI subgroup of the group generated by the scale, in such a way that tempered versions of each are equivalent. This is particularly useful for analyzing planar temperaments, as the projection can then be viewed in lattice form by Scala's "lattice" or "lattice and player" command.
An example of a projection pair is 7 225/32, which when applied by Scala's "project" to a 7-limit scale produces a 5-limit scale, which when tempered by marvel (225/224) temperament gives exactly the same result as the original scale does when also tempered. This can be thought of as marvel temperament replacing 7 by 225/32.
More than one such pair may be required to reduce to the desired subgroup; for instance 7 225/32, 11 4096/375 reduces an 11-limit JI scale to a 5-limit JI scale equivalent under (undecimal) marvel. This can happen even when only one comma is involved (codimension one temperaments). For instance, to project a 7-limit scale in the hemimean (3136/3125) reduction to the 2.5.7 subgroup requires 5 3136/625, 7 68841472/9765625.
Many projection pairs are given on the pages for various planar temperaments. When no subgroup is indicated, the default 2.3.5 5-limit subgroup is presumed. These lists of pairs can be copied and pasted into Scala and applied to any suitable JI scale.
List of projection pairs
5-limit
| Temperament | Associated comma | Projection pair | Target subgroup |
|---|---|---|---|
| Negri | 16875/16384 | 3 50625/16384, 5 16384/3375 |
2.15 |
| Porcupine | 250/243 | 3 729/250, 5 59049/12500 |
2.9/5 |
| Magic | 3125/3072 | 3 3125/1024 |
2.5 |
| Tetracot | 20000/19683 | 3 20000/6561, 5 2000000000/387420489 |
2.9/5 |
| Meantone | 81/80 | 5 81/16 |
2.3 |
| Würschmidt | 393216/390625 | 3 390625/131072 |
2.5 |
| Orson | 2109375/2097152 | 3 13348388671875/4398046511104, 5 2097152/421875 |
2.75 |
| Hanson | 15625/15552 | 3 46656/15625, 5 15552/3125 |
2.5/3 |
| Helmholtz | 32805/32768 | 5 32768/6561 |
2.3 |
7-limit
| Temperament | Associated comma | Projection pair | Target subgroup |
|---|---|---|---|
| Keegic | 1029/1000 | 3 1000/343 |
2.5.7 |
| Mint | 36/35 | 7 36/5 |
2.3.5 |
| Avicennmic | 525/512 | 7 512/75 |
2.3.5 |
| Semaphoresmic | 49/48 | 3 49/16 |
2.5.7 |
| Sengic | 686/675 | 5 3375/686, 7 675/98 |
2.3.7/5 |
| Archytas | 64/63 | 7 64/9 |
2.3.5 |
| Blackjackismic | 854296875/843308032 | 5 843308032/170859375, 7 5903156224/854296875 |
2.3.7/5 |
| Squalentine | 64827/64000 | 5 320000/64827, 7 64000/9261 |
2.3.7/5 |
| Keemic | 875/864 | 7 864/125 |
2.3.5 |
| Gariboh | 3125/3087 | 5 15625/3087, 7 9765625/1361367 |
2.3.25/7 |
| Nuwell | 2430/2401 | 5 2401/486 |
2.3.7 |
| Trimyna | 50421/50000 | 3 50000/16807 |
2.5.7 |
| Sensamagic | 245/243 | 5 243/49 |
2.3.7 |
| Starling | 126/125 | 7 125/18 |
2.3.5 |
| Octagar | 4000/3969 | 5 3969/800, 7 27783/4000 |
2.3.7/5 |
| Orwellismic | 1728/1715 | 5 1728/343 |
2.3.7 |
| Gamelismic | 1029/1024 | 3 1024/343 |
2.5.7 |
| Marvel | 225/224 | 7 225/32 |
2.3.5 |
| Cataharry | 19683/19600 | 3 19600/6561, 7 1033052339200000000/150094635296999121 |
2.5.81/7 |
| Mirkwai | 16875/16807 | 5 84375/16807, 7 16875/2401 |
2.3.7/5 |
| Hemimage | 10976/10935 | 5 10976/2187 |
2.3.7 |
| Hemimean | 3136/3125 | 5 3136/625, 7 68841472/9765625 |
2.3.25/7 |
| Hemifamity | 5120/5103 | 7 5120/729 |
2.3.5 |
| Porwell | 6144/6125 | 3 6125/2048 |
2.5.7 |
| Garischismic | 33554432/33480783 | 7 33554432/4782969 |
2.3.5 |
| Wadismic | 201768035/201326592 | 5 201326592/40353607 |
2.3.7 |
| Quasiorwellismic | 29360128/29296875 | 7 29296875/4194304 |
2.3.5 |
| Horwell | 65625/65536 | 7 65536/9375 |
2.3.5 |
| Metric | 703125/702464 | 5 702464/140625, 7 3454189699072/494384765625 |
2.3.25/7 |
| Wizmic | 420175/419904 | 5 882735153125/176319369216, 7 419904/60025 |
2.3.245 |
| Breed | 2401/2400 | 3 2401/800 |
2.5.7 |
| Ragismic | 4375/4374 | 7 4374/625 |
2.3.5 |
11-limit
| Temperament | Associated comma | Projection pair | Target subgroup |
|---|---|---|---|
| 33/32 | 11 32/3 |
2.3.5.7 | |
| Cake | 45/44 | 11 45/4 |
2.3.5.7 |
| 55/54 | 11 54/5 |
2.3.5.7 | |
| 56/55 | 11 56/5 |
2.3.5.7 | |
| Frostmic | 245/242 | 5 242/49 |
2.3.7.11 |
| Mothwellsmic | 99/98 | 11 98/9 |
2.3.5.7 |
| Ptolemismic | 100/99 | 11 100/9 |
2.3.5.7 |
| Biyatismic | 121/120 | 5 121/24 |
2.3.7.11 |
| 1331/1323 | 7 9261/1331, 11 1323/121 |
2.3.5.11/7 | |
| Valinorsmic | 176/175 | 11 175/16 |
2.3.5.7 |
| Pentacircle | 896/891 | 11 896/81 |
2.3.5.7 |
| 4375/4356 | 7 4356/625 |
2.3.5.11 | |
| Semicanousmic | 14641/14580 | 5 14641/2916 |
2.3.7.11 |
| Rastmic | 243/242 | 3 242/81, 11 644204/59049 |
2.5.7.11/9 |
| 3388/3375 | 7 3375/484 |
2.3.5.11 | |
| Keenanismic | 385/384 | 11 384/35 |
2.3.5.7 |
| Trimitone | 8019/8000 | 11 8000/729 |
2.3.5.7 |
| Werckismic | 441/440 | 11 441/40 |
2.3.5.7 |
| 1375/1372 | 11 1372/125 |
2.3.5.7 | |
| 6250/6237 | 11 6250/567 |
2.3.5.7 | |
| Swetismic | 540/539 | 11 540/49 |
2.3.5.7 |
| 4000/3993 | 3 4000/1331 |
2.5.7.11 | |
| Symbiotic | 19712/19683 | 11 19683/1792 |
2.3.5.7 |
| 5632/5625 | 11 5625/512 |
2.3.5.7 | |
| Argyric | 41503/41472 | 7 41472/5929, 11 456533/41472 |
2.3.5.77 |
| Lehmerismic | 3025/3024 | 7 3025/432 |
2.3.5.11 |